J Sign Process SystDOI 10.1007/s11265-008-0277-4

Using Heterogeneity to Enhance RandomWalk-based Queries

Marco Zuniga · Chen Avin · Bhaskar Krishnamachari

Received: 1 October 2007 / Revised: 24 July 2008 / Accepted: 3 September 2008© 2008 Springer Science + Business Media, LLC. Manufactured in The United States

Abstract It is a well-known property of random walksthat nodes with higher degree are visited more fre-quently. Based on this property, we propose the useof cluster-heads (high-degree nodes) together with asimple push-pull mechanism to enhance the perfor-mance of random walk-based querying: events arepushed towards high-degree nodes (cluster-heads) andpulled from the cluster-heads by a random-walk origi-nated at the sink. Following this simple mechanism, weshow that having even a small percentage of cluster-heads (degree-heterogeneity) can provide significantimprovements in query performance. For linear topolo-gies, we use connections between random walks andelectrical resistances to prove that placing uniformly afraction of 4/5k cluster-heads (where 2k is the degree

This work was supported in part by NSF through grantsnumbered CNS-0325875, CNS-0347621, CNS-0435505,and CCF-0430061.

The work was done while the second author was a PostDocat USC

M. Zuniga (B) · B. KrishnamachariDepartment of Electrical Engineering—Systems,University of Southern California, Los Angeles,CA 90089-0781 USAe-mail: marcozun@usc.edu

B. Krishnamacharie-mail: bkrishna@usc.edu

C. AvinDepartment of Communication Systems Engineering,Ben Gurion University of The Negev,Beer Sheva, 84105, Israele-mail: avin@cse.bgu.ac.il

of each cluster-head), can reduce querying costs from�(n2) to �(n2/k2), an improvement of �(k2). For morerealistic two-dimensional topologies, we use Markovchain analysis and simulations to show a similar trend—using about 10% of the nodes as cluster-heads pro-vides a query cost improvement between 30% and 70%depending on the coverage of the high-degree nodes.

Keywords Wireless sensor networks · Querying ·Random walk · Electrical resistance · Markov chains

1 Introduction

An important approach for querying in unstructuredsystems is the use of random walks. This approach isgaining popularity in the networking community sincerandom walks are intuitively simple—nodes are visitedsequentially in a random order with successive nodesbeing neighbors in the graph [21–24]. There is alsoa significant body of theoretical literature on randomwalks as querying mechanisms [9–11]. However, whilethis literature provides much insight into the scalingbehavior of random walks on simple classes of deter-ministic graphs (such as 2D Torii), a major property ofreal-life networks, degree-heterogeneity, was left out ofdiscussion.

Heterogeneity on the degree distribution is a highlylikely characteristic in real large-scale wireless systems,such as sensor networks, for a number of reasons.First, random deployments are inherently non-regulargraphs. Second, empirical studies [2, 30] have revealedthat hardware variance on the sensitivity and out-put power of radios lead to nodes with significantlyhigher degree than the average (cluster-heads). Third,

J Sign Process Syst

some works [3] have recently proposed that for wire-less sensor networks to scale, heterogenous networksconsisting of highly capable and low capable devices arerequired.

In this work we propose the use of a push-pull mech-anism to exploit the heterogeneity of the underlyingcommunication graph to enhance the performance ofrandom-walk-based queries. We take advantage of awell known property of the simple random walks: itsstationary distribution π(v) = d(v)/2m, where d(v) de-notes the degree of node v and m the number of edgesin the graph. This means that nodes with higher degreeare visited more frequently by the random walk. Thisquerying mechanism is particulary suitable for one-shotqueries in scenarios where data is replicated. Examplesof possible applications are habitat monitoring: Hasanimal x been detected? and environmental monitor-ing: Is the temperature higher than t anywhere in thenetwork?

The idea we use is intuitively simple, events arepushed towards high-degree nodes (cluster-heads) andpulled from the cluster-heads by performing a random-walk-based query originated at the sink. In this scenariosome important questions arise: What is the impact ofheterogeneity on performance? and How much hetero-geneity is needed?

We use two theoretical tools to explore these ques-tions. The first tool is used for 1-D (linear) topologies.It is based on the direct connection between a randomwalk on a graph and the resistance of the electricalnetwork obtained from the graph by viewing each edgeas a unit resistor [28, 29]. Using this tool, we bound theinfluence of cluster-heads on the query cost. The secondtool is applied on 2-D topologies and it is based onabsorption states of Markov chains, we use this tool toobtain the expected number of steps (cost) in a query.

The main contribution of this work is to show thatheterogeneity, together with our simple push-pull al-gorithm, allows random-walk-based queries to enhancetheir performance. In particular, the important resultwe obtain is that for a line topology where cluster-heads have a coverage k (cover k nodes to the rightand k nodes to the left) and are uniformly distributed(evenly spaced), a fraction of 4

5k nodes being cluster-heads can offer a reduction in query cost of �(k2).In intuitive terms this translates to requiring less than10% of the nodes being cluster-heads to obtain twoorders of magnitude improvement in query cost. Also,through numerical analysis, we present results showingthat in 2D networks also a small percentage of nodesbeing cluster-heads (∼ 10%) can lead to significantimprovements in performance (between 30% and 70%depending on the coverage of the high-degree nodes).

2 Enhancing Random Walks for Heterogeneity

In this section we present the event-push query-pullalgorithm used in our work. We consider a networkwhere two type of nodes are available: i) nodes withlimited communication capability (low degree) and ii)nodes with higher communication capabilities (highdegree, cluster-heads). Our focus is on infrastructure-less networks (no location nor GPS capabilities), wherenodes are able to communicate only with their neigh-bors, and they are aware of their neighbors degree (ifneighbors are cluster-heads or not).

Our query mechanism is built from two parts: firstan event ε is generated randomly at any node in thenetwork, and second, a random-walk-based query isissued in order to find the event. The event ε can eitherremain at the node where it was generated or move toa cluster-head (if one exists). In a network where allnodes have the same degree, the event remains at thenode where it appears. When cluster-heads are present,upon detection of an event, the event searches for thecloses node with the highest degree (local maxima) asshown in Algorithm 1.

Algorithm 1 Event forwarding in Heterogeneous Network

Require: event at node1: while node is not a cluster-head do2: if there is a neighbor with degree( ) degree(

then3: forward to the neighbor with the highest degree;

break ties uniformly at random among candidates4: else5: forward to a random neighbor6: end if7: end while

In order to find the event, a query is issued througha predefined sink node. The query follows a simplerandom walk where the next node is chosen uniformlyfrom the neighbors, until it finds the event ε.

In the scenario described above there will be twocosts: i) the cost of moving the event to a cluster-head,Cevent and the cost of the query (i.e random walk) tofind the event, Cquery. The total cost is the sum of both:Ctotal = Cevent + Cquery.

Denoting L as the number of low-degree nodes andH as the number of high-degree nodes, we are inter-ested in the following questions: How does Ctotal varywith the ratio H

H+L ? Is there a value of HH+L that will

reduce Ctotal significantly?

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3 Analytical Results

In this section we focus on line topologies and we areinterested in analyzing the impact of cluster-heads onthe total cost incurred in discovering an event (Ctotal).The main result is as follows:

Theorem 1 Consider a line topology with (n + 1) nodesand a sink positioned at one of the extremes of the line.When cluster-heads with a 2k-degree are uniformly dis-tributed on the line, the first local minima for the hittingtime is obtained when the fraction of high-degree nodesis 4

5k . And for this fraction, the hitting time is reducedfrom �(n2) to �(n2/k2) (a �(k2) improvement).

This result is obtained using the relationship betweenrandom walks and electrical circuits. The remainder ofthe section is dedicated to the proof of this theorem,and Table 1 presents the notation used on the proof.

3.1 Background on Random Walks and ResistanceMethods

For a graph G(V, E) where |V| is the number of nodesand |E| is the number of edges, the following notationwill be used. An element of V or E is represented bya lowercase, a subset or array is represented by a boldlowercase and the complement of a set or element willbe denoted with an upper-bar, for example e representsan element, e an array (subset) and e and e representthe complements of e and e, respectively.

The hitting time huv is the expected time taken by asimple random walk starting at u to reach v for the firsttime [10].

For a source u ∈ V and the subset of cluster-headnodes v ⊆ V, the average hitting time from u to v is:

huv =

∑

v∈v

huv

|v| (1)

The commute time Cuv is defined as the expectedtime taken by a random walk starting at u to reachv and come back to u. Note that by definition Cuv =huv + hvu, but in general huv �= hvu. In a seminal work,Doyle and Snell [28] explored the connection betweena random walk on a graph G and the resistance of an

Table 1 Notation.

(n + 1) Number of nodesk Cluster coveraged Inter cluster-heads distanceα Overlaping of cluster-heads coverage (region 2)(s + 1) Number of clusters

electrical network obtained from G by viewing eachedge as a unit resistor. In [29], Chandra et al. extendedthis work and proved the following equality that relatesthe commute time Cuv and the effective resistance Ruv

of the electrical network of G :

Cuv = 2mRuv (2)

Where m is the number of edges in the graph. Noticethat in case of symmetry1 huv = hvu, which implies thatthe commute time is two times the hitting time. We willuse this property in the analysis of the hitting time forline topologies.

3.2 Parameters of Line Topology

We consider the undirected graph L(n, k, d) = G(V, E)

with the following parameters. The set of nodes isV = v0, v1, . . . vn, since the index goes from 0 to n, thenumber of nodes is |V| = n + 1, we will also denote nas the number of edges when no cluster-head has beenadded (i.e. when nodes communicate only with theirimmediate right and left neighbors). In addition to theinitial n edges, each cluster-head has edges to all nodeswhich are less than or equal to k nodes away from iton the line. d is the inter cluster-head distance, hence,a node vi is a cluster-head iff i mod d = 0. The set ofedges E for L(n, k, d) is defined as follows:

E = {viv j | (i mod d = 0 and |i− j| ≤ k) or |i − j|=1

}

We will consider the case where n � k > 1. Since weare interested in the limit n → ∞ we will consider onlythe cases where n = sd, where s is an integer. Then, thetotal number of clusters is given by (s + 1).

For a given n and k we are interested in studying theimpact of d on Ctotal via the resistance, and the analysisis divided in different regions according to the intercluster-heads distance d:

regions =⎧⎨

⎩

region 1, 2k ≤ d ≤ n/2region 2, k < d < 2kregion 3, 1 ≤ d ≤ k

(3)

Figure 1 presents examples of line topologies for the3 different regions. The big circles denote cluster-headsand the dashed lines their coverage k. Notice that k isconstant for all three topologies.

Later in this section we will show that Ctotal dependsmainly on Cquery, specially for d < 2(k + 1) where Cevent

will be shown to be less than 1. For this reason, theresistance analysis will be focused on Cquery.

1Symmetry refers to a graph G′, where we name u as v and viceversa, is isomorphic to G.

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Region 3

Region 2

Region 1 δ1

δ2

Figure 1 Examples of line topologies. The big circles denotecluster-heads and the dashed lines their coverage k, all nodeswithin k hops from the cluster-head are its neighbors.

For the different regions we will first obtain expres-sions for the number of edges m and the effective resis-tance R between the nodes at the extremes of the line.Then, in Subsections 3.3 and 3.4 we use these expres-sions, together with symmetry, to obtain the maximumhitting time and average query cost, Cquery.

1. Analysis of region 1 : Recalling that the number ofnodes in L is (n + 1), and that when no clusters arepresent the initial number of edges is n. Besidesn, each cluster-head contributes with 2(k − 1) newedges. Then the total number of edges is given by:

m1 = n + 2(k − 1)s

= n + 2(k − 1)n/d

= nd

(d + 2(k − 1)) (4)

The coverage on each side of a cluster-head can berepresented by an effective resistance r(k) (Fig. 1). r(k)

can be derived based on k using techniques to reduceresistors in parallel and series:

k = 2 ⇒ r(2) = 2/3

k = 3 ⇒ r(3) = 5/8

k = 4 ⇒ r(4) = 13/21

It can be derived that the numerator and denomina-tor follows the fibonacci series f ib(·), hence:

r(k) = f ib(2k − 1)

f ib(2k)(5)

Since every cluster-head covers its right and leftk-neighbors and the extreme vertices cover only one

side, the effective resistance R1 between the extremesof L is given by:

R1 = (2r(k) + d − 2k)s

= (2r(k) + d − 2k)n/d (6)

Finally, using Eq. 2 and symmetry, the hitting timefor nodes at the extreme of the line is given by:

h1 =(n

d

)2(d + 2(k − 1)) (d + 2(r(k) − k)) (7)

2. Analysis of region 2 : The number of edges is thesame as Eq. 4 since every cluster-head that is addedbrings 2(k − 1) new edges. However, the effectiveresistance between 2 cluster-heads is different. Inthis subsection we provide upper and lower boundsfor this resistance.

Letting αk be the overlap between the coverageof two neighboring cluster-heads, where 0 < α < 1,then d = k(2 − α). The resistance circuit is equivalentto the one shown in Fig. 2, where r(x) = r(k(1 − α))

(the effective resistance for the non-overlapping part).Given that r() converges to the inverse of the goldenratio, 1/2 < r(x) < 1, further, considering Rayleigh’smonotonicity law we can provide upper and lowerbounds for the resistance as shown in Fig. 2:

2

αk + 2< r2 <

2

αk + 1

2

αk + 2s < R2 <

2

αk + 1s

2

αk + 2

nd

< R2 <2

αk + 1

nd

(8)

Figure 2 Resistance calculation for regions 2 and 3.

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Finally, the hitting time is bounded by:

2(n

d

)2(

4k − kα − 2

αk + 2

)< h2

< 2(n

d

)2(

4k − kα − 2

αk + 1

)(9)

3. Analysis of region 3 : For this region, we will ana-lyze only the point where d = k.

Contrary to the case of regions 1 and 2, neigh-boring cluster-heads have an edge connecting eachother, hence each cluster-head brings (2(k − 1) − 1)new edges. And the total number of edges is given by:

m3 = n + (2k − 3)s

= n + (2k − 3)n/d

= 3nk

(k − 1) (10)

The resistance between two cluster-heads can betransformed to an equivalent circuit as shown in Fig. 2,which leads to an equivalent resistance of 2

k+1 betweentwo neighboring cluster-heads. Hence, the effectiveresistance between the extremes of the line is given by:

R3 = 2

k + 1s

= 2

k + 1

nk

(11)

Finally, the hitting time for d = k is given by:

h3(d = k) = 6(n

k

)2 k − 1

k + 1(12)

Tight analytical results for the case d < k are harderto obtain, but we can show the following lower bound:2

h3 ≥ 2n2

k2 + 3k(13)

3.3 Local Minimum for Maximum Hitting Time

In this section we analyze the hitting time betweenthe sink (v0) and the last cluster-head on the line (vn).Table 2 shows the minimum value of h1, h2 and h3. Inregion 3, we analyzed only one point (d = k) , hence,for region 3 the value in Table 2 is the one obtainedin Eq. 12. For region 1, the minimum value is obtainedfor d = 2k, we further assume a lower bound on h1 bysetting r(k) = 0.5.

2The proof of this result is not shown due to lack of space.However, it does not affect in anyway the result in Theorem 1.It is presented only on the interest of completeness.

Table 2 Minimum maximum and minimum average hitting timesper region.

Maximum Averagehitting time (h) hitting time (Cquery)[ Subsection 3.3 ] [ Subsection 3.4 ]

Region 1(n

k

)2(

k − 1

2

)(2k − 1)n(n + k)

6k2

(lower bound)

Region 2 2

(4n5k

)2 (13k − 8

3k + 4

)4n(13k − 8)(8n + 5k)

3(3k + 4)(5k)2

(upper bound)

Region 3 6(n

k

)2(

k − 1

k + 1

)(k − 1)(2n + k)n

(k + 1)k2

In the case of region 2, the hitting time depends onthe overlapping α. Even though αk should take onlyinteger values, we assume α to be real in order todifferentiate h2 (Eq. 9) and obtain the value of α thatminimizes h2. The values of α for the upper and lowerbounds are given by (αU and αL):

αU = 12nk − 7n + 4k − 16k2

2(2n − 4k + 1)k

−√

80n2k2−88n2k+96nk2−128nk3+17n2+48nk−16n2(2n − 4k + 1)k

(14)

αL = −8k2+6nk−4n−2√−8k3n+5n2k2−4n2k+8nk

2k(n − 2k)

(15)

For large n and k, αU = αL = 0.7639, hence αopt ≈ 34 ,

Table 2 shows the upper bound of h2 for αopt.From Table 2 we observe that h1 = �(n2/k), and h2

and h3 are �(n2/k2), hence for large k the minimumis either on h2 or h3. Since n and k are given, we cancompare directly upper bound of h2 and h3 given inTable 2, which leads to:

limk→∞

h3

h2= 1.0817 (16)

Hence, the first local minima is in region 2 and theinter cluster-head distance that attains this minimum isd = 5

4 k, which corresponds to a ratio � between high-degree nodes and the total number of nodes:

� =(

4n5k

)(1

n + 1

)≈ 4

5k(17)

It is important to notice that the global minima mightbe in region 3, however, the reduction obtained bymoving from the first local minima to the global minimais not significant, as it will be shown numerically in thenext section. Furthermore, as presented in Eq. 13 thecost in region 3 is the same order as in region 2.

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3.4 Local Minimum for Expected Hitting Time

For regions 1, 2 and region 3 (when d = k), cluster-head i is visited only after cluster-head i − 1 has beenvisited. This property allows us to discard all nodesbeyond a given cluster-head i when we are interestedin obtaining the hitting time to i. Hence, for cluster-head i we can consider a new line topology, which isa shorter version of the original one, where the sinkis at one extreme and cluster-head i is at the end ofthe new line. This behavior allows us to use directlythe expressions derived in the previous subsectionswith the only change to be made in the initial numberof nodes n and edges m.

Recalling that the distance set between the sinkand cluster-heads is given by {0, d, 2d, 3d...(s − 1)d, sd}.Cquery is the average hitting time (E[hi]) to all clustersand for region 1 is given by:

E[h1] =

s∑

i=0

h1(n = id)

s + 1

=

s∑

i=1

(nd

)2(d + 2(k − 1)) (d + 2(r(k) − k))

s + 1

= (d + 2(k − 1)) (2r(k) + d − 2k)

s + 1

s∑

i=1

i2

= (d + 2(k − 1)) (2r(k) + d − 2k)s(2s + 1)

6(18)

For Region 2, using the upper bound of Eq. 9 andrecalling that d = k(2 − α), the expected hitting time isgiven by:

E[h2] =

s∑

i=0

h2(n = id)

s + 1

=

s∑

i=1

2(n

d

)2(

4k − kα − 2

αk + 1

)

s + 1

= 2(4k − kα − 2)

(αk + 1)(s + 1)

s∑

i=1

i2

= n(2n + 1)(4k − kα − 2)

3(αk + 1)(k(2 − α))2(19)

The derivative of this equation with respect to α

leads to an expression that is considerably more compli-cated than Eq. 14 and it is not presented. However, the

results are the same as the obtained for the maximumhitting time, i.e. αU = αL = 0.7639 and αopt ≈ 3

4 . Theclosed-form expressions and values can be easily ob-tained by using mathematical software such as Matlabor Mathematica.

For Region 3, when d = k, the expected hitting timeis given by:

E[h3(d = k)] =

s∑

i=0

6(n

k

)2 k − 1

k + 1

s + 1

= 6(k − 1)

(s + 1)(k + 1)

s∑

i=0

i2

= k − 1

k + 1s(2s + 1) (20)

Table 2 also presents the minimum value of Cquery

for the 3 regions: d = 2k, r(k) = 0.5 for region 1 (lowerbound), d = 5

4 k for region 2 (upper bound) and d = kfor region 3 (only one point analyzed). The compar-isons lead to the same result as the ones obtained forthe maximum hitting time: the order of h1 is greaterthan the order of h2 and h3, and as n and k goes toinfinity the ratio of h3 over h2 goes to 1.0817.

Now we have all the elements to prove Theorem 1

Proof of Theorem 1 The optimization of the maximumand average hitting times (Eqs. 9 and 19) leads tod = 5

4 k, using Table 2 we proved that the first localminima occurs in region 2 for both the maximum andaverage hitting times, and that their cost are �(n2/k2).It is known that random walks on regular line topolo-gies have a cost of �(n2) [9], hence, a line topol-ogy L(n, k, d) with d = 5

4 k leads to a cost reductionof �(k2) for both the maximum and average hittingtimes. ��

4 Numerical and Experimental Results

In this section we use Markov numerical methods andsimulations to study the impact of heterogeneity onthe performance of random walk-based queries. First,we briefly introduce the method of using absorptionstates to obtain hitting times. Then, we present nu-merical results on: (a) line topologies (validating theanalytical contributions of Section 3) and (b) regulargrids and random geometric graphs. Finally, we presentsimulation results on realistic graphs for wireless sensornetworks.

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4.1 Background on Hitting Times in Markov Chains

Given a graph G(V, E), a random walk on the graphcan be defined by a Markov chain M, where M rep-resents the transition probability matrix. The notationused for elements and subsets of V and E is the sameas the one presented in Subsection 3.2.

Let us consider that the event is in node e ∈ Vand the sink is in node s ∈ V. As presented in [27],absorption states can be used to obtain hitting times.Let Me be the matrix resulting from deleting the rowand column corresponding to e in M, and let Qe be:

Qe = (I − Me)−11 (21)

Where I is the identity matrix and 1 is a columnvector of ones. Qe is an array representing the expectedhitting time from each node to e (except e). Hence,in our case the hitting time from s to e is given by:hse = Qe(s), where Qe(s) represents the s’th element ofthe array.

4.2 Line Topologies

For a given line topology L(n, k, d) with transitionprobability matrix M, where the sink is the extreme leftvertices v0. Clusters-heads are positioned at vid wherei = 0, 1, ..., s.

The hitting time from v0 to a specific cluster-head isQid(v0), where Qid is given by:

Qid = (I − Mid)−11 (22)

And Cquery, the average hitting time over all cluster-heads, is given by:

Cquery = d2n

(Q0(v0) + Qsd(v0)

) + dn

s−1∑

i=1

Qid(v0)

= d2n

(Qsd(v0)) + dn

s−1∑

i=1

Qid(v0) (23)

In the previous equation all cluster-heads have thesame weight except for the extreme ones (v0 and vsd),this is due to the fact that at the extremes, the expectednumber of stored events is half of those stored at theintermediate cluster-heads. However, for large numberof cluster-heads the weight can be considered similarand:

Cquery ≈

s∑

i=1

Qid(v0)

s + 1(24)

Cevent will be derived according to the regions definedin Eq. 3 and Algorithm 1. There are s + 1 cluster-heads

for which there is no need to move the event, hence thecost is zero.

When d < 2(k + 1), all nodes will be directly con-nected to a cluster-head and hence Cevent = n−s

n+1 . How-ever, when d ≥ 2(k + 1) (most of region 1), there willbe orphan nodes between any pair of consecutivecluster-heads. Due to symmetry, the cost will be thesame for any subset of nodes between any two neigh-boring cluster-heads and for simplicity we considercluster-heads v0 and vd. For these clusters, located at0 and d, nodes between (k + 1) and (d − k − 1) areorphan. Let us define a = (k + 1), b = (d − k − 1) andM(a:b) as the M’s sub-matrix which includes only therows and columns between a and b . For Ma:b, Qa:b =(I − Ma:b )−11 is the array containing the expectednumber of steps of each orphan node to reach the clos-est node directly connected to a cluster-head. Hence,the cost of moving orphan nodes between a and b to acluster-head is given by:

Corphan =∑

i∈Q

(Qa:b (i) + 1) (25)

Where the constant 1 represents the cost of movingthe event from a node connected to a cluster-head tothe cluster-head.

Finally, recalling that (n + 1) is the total number ofnodes in L, Cevent is given by:

Cevent =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2k(s + 1) + sCorphan

n + 1, 2(k + 1) ≤ d ≤ n/2

n − sn + 1

, 2k ≤ d < 2(k + 1)

n − sn + 1

, region 2

n − sn + 1

, region 3

(26)

Figure 3 shows Cquery, Cevent and Ctotal vs the numberof clusters for a line topology with 121 nodes anddifferent values of k. The solid lines represent the costobtained using Markov numerical analysis (Eqs. 24and 26), and the dotted lines represent simulationresults, it can be observed that the Markov methodprovides an accurate representation of the cost. Also itmust be noted that the query cost accounts for most ofthe total cost, which validates the focus of the analyticalsection on Cquery.

Figures 4 and 5 compare the maximum and expectedhitting time between the Markov analysis (full lines)and the expressions obtained through the resistancemethod (dotted lines) for a line topology with 121nodes and values of k ranging from 5 to 10. The dottedlines show that the bounds get tighter for higher values

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101

102

100

101

102

103

104

number of clusters

cost

(st

eps)

Cost of Event

κ = 6

κ = 10κ = 8

region 1

region 2

region 3

MarkovSimulations

101

102

102

103

104

cost

(st

eps)

Cost of Query

number of clusters

κ = 6

κ = 10

κ = 8

region 2

region 1

region 3

MarkovSimulations

100

101

102

102

103

104

cost

(st

eps)

number of clusters

Total Cost

κ = 6

κ = 10

κ = 8

region 1

region 2

region 3

MarkovSimulations

a

b

c

� Figure 3 Cquery, Cevent and Ctotal vs the number of clusters for aline topology with 121 nodes and different values of k (6, 8 and10). The solid lines represent the cost obtained using Markov nu-merical analysis and the dotted lines represent simulation results(a–c).

of k in both figures. The figures also shows a linewith circle markers depicting the number of clustersrequired to reach the first local minima according to ouranalysis (Eq. 14), which supports the results presentedin Theorem 1. It is important to notice that the analyti-cal values for the number of clusters are not necessarilyintegers, and hence, they may not match exactly thenumerical ones, specially for low values of k. Howeverfor large values of k and n, the floor or ceiling of theanalytical value will not incur in significant differences.

It is also important to notice that the first cluster-heads added leads to a significant cost reduction (region1 and part of region 2), adding even more high-degreenodes beyond this point provides diminishing returnsor in some cases even degrades the performance. Anintuitive explanation for the diminishing return phe-nomenon is that when a small percentage of nodes arecluster-heads most of the events will be pushed andpulled from these few nodes, hence, the random walkwill not only need to visit a small number of high-cluster nodes to retrieve most of the queries, but itwill visit them frequently. However, when more cluster-heads are added these desirable properties vanish, the

101

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103

104

Maximum Hitting Time

number of clusters

cost

(st

eps)

markov

analytical first local minima

κ = 5

κ = 10

κ = 6 κ = 7

κ = 8 κ = 9

region 2

resistancebounds

Figure 4 Maximum hitting time for a line topology with 121nodes and values k ranging from 5 to 10. The full lines representMarkov numerical values and the dotted lines the analytical re-sults through resistance methods. The line with the circle markersrepresent the analytical values of the number of cluster heads forthe first local minima.

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101

102

102

103

104

Expected Hitting Time

number of clusters

cost

(st

eps)

κ = 5

κ = 10

markov

resistancebounds

analytical first local minima

κ = 9 κ = 8

κ = 6 κ = 7

region 2

Figure 5 Cquery (expected hitting time) for a line topologywith 121 nodes and values k ranging from 5 to 10. The fulllines represent Markov numerical values and the dotted lines theanalytical results through resistance methods. The line with thecircle markers represent the analytical values of the number ofcluster heads for the first local minima.

random walk needs to visit more nodes to resolve thequeries and these nodes have a lower probability ofbeing visited. On the extreme cases, when all nodes areeither low-degree or high degree, no event is pushedand all nodes have the same probability of being visited.

4.3 Regular Grids and Random Geometric Graphs

Grids and random geometric graphs are common mod-els to study various properties and protocols for wire-less systems. In the previous section, we showed thatin line topologies the addition of cluster-heads cangreatly reduce the cost of random-walk-based queries,do cluster-heads have the same significant effect on 2-dimensional topologies?

1) Grids: We assume that the number of cluster-headsis a perfect square and they are uniformly distrib-uted on the grid, i.e. the grid is divided in the samenumber of cells as the number of cluster-heads, andeach cluster-head is positioned in the node at thecenter of each cell.

According to the algorithm presented in 1, eventsappearing in a cluster-head has a cost of 0, eventsappearing in nodes directly connected to a cluster-headhave a cost of 1 and events appearing in orphan nodesperform a simple random walk until it hits a node that

100

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cost

(st

eps)

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κ = 2

κ = 6

κ = 3

κ = 5 κ = 4

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cost

(st

eps)

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κ = 2

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κ = 4

κ = 5

10_

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_1

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101

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103

number of clusters

cost

(st

eps)

Total Cost

κ = 2

κ = 6

κ = 3

κ = 4

κ = 5

all nodeslow_degree

all nodeshigh_degree

a

b

c

Figure 6 a Cevent, b Cquery and c Ctotal for a grid topology with169 nodes and values of k ranging from 2 to 6.

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is directly connected to a cluster-head. Denoting: (a) Mas the transitional probability matrix, (b) w ⊆ V as thesubset of vertices containing all cluster-heads and allthe nodes directly connected to them and (c) w ⊆ V asits complement; we define Mw as the sub-matrix whereall the rows and columns of the vertices in w have beenremoved. Hence, the hitting time for orphan nodes isgiven by:

hww =|w|∑

i=0

(Qw(i) + 1) (27)

And the event cost (Cevent) for the grid topology isgiven by:

Cevent = (|w| − (s + 1)) + hsw

n(28)

In our analysis the sink is located at the bottom-left corner of the grid (v0). The query cost Cquery is theaverage hitting time from the sink to the set of cluster-heads and it is given by combining Eqs. 1 and 21.

Figure 6 presents results for Cevent, Cquery and Ctotal

((A), (B) and (C) respectively) for 169 nodes deployedon a 2D grid. The y axis represent the cost and the xaxis the number of clusters. There are two importantobservations: i) contrary to the line topology, the casewhen all nodes are high-degree perform significantlyworse, ii) similar to the line topology, the first cluster-heads account for most of the savings (greater than30%) and the higher k the higher the savings. Also notethat, as k increases, the case where all nodes are high-degree approaches a complete graph.

Another important difference with respect to the linetopology is that the event cost plays a more significantrole in the total cost, while the reduction in query costis not as significant. The lower reduction in query costmay be due to several reasons, one of them could bethat contrary to the resistance calculation done in lines,where edges beyond the cluster-head of interest werediscarded; in grids we can not eliminate edges beyond agiven cluster, hence according to Eq. 2, considering alledges for all clusters would lead to query costs that aremore similar in grid deployments.

2) Random Geometric Graphs: The procedure forgetting event and query costs in random geomet-ric graphs is the same as for grids (Eq. 28, anda combination of Eqs. 1 and 21). However, theinteresting case in random geometric graphs is thateven when only low-degree nodes are deployedthere are some inherent cluster-heads due to somefavorable geographical position. We further en-hance these natural cluster-heads by increasingtheir transmission range.

Table 3 Random geometric graphs.

Transmission Clustering No clustering Savings (%)range

0.12 679.7 833.0 18.40.18 414.2 296.2 −39.80.24 171.4 225.0 23.80.30 88.0 202.7 56.60.36 52.9 193.5 72.7

According to the algorithm presented in 1, in thesescenarios the event moves in a greedy way towardsthe local cluster-head. Table 3 presents results for 169nodes deployed randomly on a 1x1 square area. Theresults are the average over 50 runs. The initial radius is0.12 which for this density gives a network connectivityprobability of ≈ 0.5. The table has two columns named“clustering” and “no clustering”.

In the “no clustering” scenarios, events are notpushed towards high-degree nodes, they stay in thenodes where they appear (Algorithm 1 is not used).And the communication coverage of all nodes is set tothe value given in the “transmission range” column (noheterogeneity).

In the “clustering” scenarios, events are pushed to-wards high-degree nodes (Algorithm 1 is used). Also,nodes start with the same transmission range (0.12).But some nodes, that due to the random deploymenthave a higher degree than their neighbors, increasetheir communication coverage to the value given inthe “transmission range” column (increased hetero-geneity). Hence, the “clustering” scenarios are two-tierdeployments where Algorithm 1 is used.

We observe that in random geometric graphsclustering and our push-pull algorithm also have a sig-nificant impact on the performance of random-walk-based queries (except for r = 0.18.3) It is importantto mention that for the initial r = 0.12 approximately∼11% of the nodes end up being local clusters.

Table 4 Grid deployments in realistic environments.

Output power (dBm) Clustering No clustering Savings (%)

−14 263.1 428.7 38.6−13 211.7 370.4 42.8−12 174.6 333.4 47.6−11 152.6 311.4 51.0−10 132.5 278.1 52.3

3We are not certain about the reason for this value. Possibly, thisis an scenario where the first local minima is not as good as aglobal minima.

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Table 5 Random deployments in realistic environment.

Output power (dBm) Clustering No clustering Savings (%)

−14 367.2 557.7 34.2−13 250.4 432.9 42.2−12 243.8 380.8 36.0−11 207.9 332.9 37.6−10 169.9 294.4 42.3

4.4 Low-Power Wireless Graphs

Using the link layer model proposed in [30], we eval-uate through simulations the effectiveness of cluster-heads in realistic graphs, which are characterized bythe presence of unreliable and asymmetric links. Inorder to guarantee the survival of the random walkwe implemented a 3-way handshake protocol similarto [31].

Tables 4 and 5 present the results for grid and ran-dom deployments. The description of the columns isthe same as Table 3: the “clustering” column representsnetworks were Algorithm 1 is used and few nodes havethe communication coverage dictated in the “transmis-sion range” column, while in the “no clustering” col-umn all nodes have the same high “transmission range”and events remain in the nodes where they appear. Wecan observe that clustering plays a significant role in re-ducing the cost of random walk-based queries (between30% and 50%). On grid deployments approximately12% of the nodes are inherent cluster-heads, while onrandom deployments about 8% of nodes are cluster-heads.

5 Experiments with Motes

In this section we present some empirical results.Thirty-one (31) micaZ motes were deployed in a chaintopology, where nodes were spaced every 1 meter. The

high-degree nodes had a higher output power than thelow degree nodes. And the low degree nodes increasedtheir output power only to reply to the high degreenodes. Each mote sent 100 test packets to measure thePRR of the links.

Communication graphs were obtained for a chaintopology where: (a) all nodes had the same low outputpower, (b) all nodes had the same high output powerand (c) for a mixture of low and high output powers.When both type of nodes are present (high and low out-put powers), the high degree nodes were evenly spacedon the chain. We simulated 300 random walk basedqueries on each of these graphs. The event followedAlgorithm 1, and the query started at the left-most partof the chain (position 0) and followed a random walkuntil it hit the event. The results are presented in Fig. 7.

Figure 7a presents the nodes’ degree. Four curvesare presented: for the low-power graph, for the graphswith 1 and 5 clusters, and for the high power graph.Small horizontal arrows depict the position of the highdegree nodes. We observe that the average high-degreeis around 15 (k = 7). Figure 7b present the average costof finding the event. The x-axis represent the numberof high-degree nodes (cluster-heads) and the y-axisthe cost. 0(S) in the x-axis represents the case wherethe events in the low-power graph (0 cluster-heads)are static and remain in the nodes where they appear.31(S) represents static events for the high-power graph(all nodes are cluster heads.). 0 and 31 represent thelow-power and high-power graphs where events followAlgorithm 1.

It is interesting to observe that by solely usingAlgorithm 1, without the presence of cluster-heads,we can obtain significant gains in both the low-powergraph (0(S) → 0 : ∼ 25%) and the high-power graph(31(S) → 31 : ∼ 45%). It is also important to highlighttwo trends that confirm the insights obtained on ouranalytical sections: (a) the first clusters account for most

Figure 7 Empirical studyof random walks ondegree-heterogeneousgraphs, a presents the degreeof the nodes, b presents thequery cost for graphs withdifferent number of clusters.

5 10 15 20 25 300

10

0

10

0

10

0

10

node position (m)

degr

ee

c = 0

c = 1

c = 5

c = 31

0(S) 0 1 2 4 5 9 14 31 31(S)0

0.1

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Rel

ativ

e C

ost

(a) (b)

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of the savings and (b) adding more cluster either leadsto diminishing returns or consumes more energy. It alsointeresting to observe that even though the degree ofthe low-power nodes is higher than 2, the ratio 4

5k leadsto approximately 3.4 cluster-heads to hit the first localminimum, which is a decent analytical prediction.

Finally, it is important to state that the empiricalstudies presented in this section are limited and areshown only as a proof-of-concept. Large scale networksneed to be tested in order to accurately asses theimpact of the contributions of this work on degree-heterogenous networks.

6 Related Work

The simplest implementation of a query disseminationprotocol for a sensor network is the basic floodingmechanism where a query message is forwarded by allnodes in the network. Flood-based queries (used, forinstance, in Directed Diffusion [4] to set up routes fromthe sources to the querying node) have the advantage ofsimplicity, and, when used in the context of continuousdata stream responses, can be justified because theircosts can be amortized over the period of the response.However, for one-shot queries, other techniques aredesired. Several researchers have studied the use of se-quential TTL-based controlled floods (expanding rings)as unstructured query mechanisms [1, 5–8]. An im-portant approach for pull-based one-shot queries inunstructured systems is the use of random walks.

Random walks on graphs have been studied math-ematically, and there is a substantial-yet-growing bodyof theoretical literature on the subject [9–11]. They arealso finding increasing use in a wide range of protocolsin the context of several networked distributed systems.For instance, they have been used in Grid-aware oper-ating systems [12], in unstructured P2P Networks [13–15], for hybrid application overlays [16], for groupmembership services in mobile ad hoc networks [17,18], for distributed model checking [19], and for indexquality determination for the world-wide web [20].

Specifically in the context of unstructured wirelesssensor networks, different variants of random-walk-based protocols have been proposed and analyzed byseveral research groups. Servetto and Barrenechea [21]proposed and analyzed the use of constrained randomwalks on a grid for performing load-balanced routingbetween two known nodes. Avin and Brito [22] haveargued that even simple random walks can be used forefficient and robust querying because they are inher-ently load-balanced and their partial cover times showgood scaling behavior. The ACQUIRE protocol [23]

provides a tunable look-ahead parameter to combinerandom walks with controlled floods and show thatsuch random-walk-based hybrids can outperform flood-ing and even expanding-ring-based approaches in thepresence of replicated data. The rumor routing algo-rithm [24] is a hybrid push-pull mechanism that advo-cates the use of multiple random walks from the eventsas well as the sinks, so that their intersection points canbe used to provide a rendezvous point. Shakkottai [25]has analyzed different variants of random-walk-basedquery mechanisms and concludes that source and sink-driven sticky-searches (similar to rumor routing) pro-vide a rapid increase of query success probability withthe number of steps. Most recently, Alanyali et al. [26]have proposed the use of random walks in energy-constrained networks to perform efficient distributedcomputation of a class of decomposable functions (use-ful in computing certain kinds of aggregates). To thebest of our knowledge, these prior studies have notinvestigated the impact of heterogeneous deploymentson the performance of random-walk-based queryingprotocols.

7 Conclusions

Our work presented an study on the impact of de-gree heterogeneity on random walk-based queries. Themain contribution of the work is showing that witha small percentage of high-degree nodes in the net-work (< 10%) and using a simple distributed push-pullmechanism, significant cost savings can be obtained —between 30% and 70% depending on the coverageof the high-degree nodes. Our work provides interest-ing theoretical results for line topologies showing thatwhen cluster-heads have a coverage k (cover k nodesto the right and left) and are uniformly distributed, afraction of 4

5k nodes being cluster-heads can improvequerying costs by �(k2) by using a simple distributedalgorithm. We also presented a limited set of resultsbased on empirical data from a test-bed of MicaZmotes.

While our study was focused on reducing query costs,it may potentially reduce delay as well. One of thedrawbacks of random walks is the significant delaythat they encounter. In our work, by minimizing therequired number of steps on the random walk we arenot only reducing the cost but also the delay. However,an accurate quantification of the savings should includespecific characteristics of the protocol used at the MAClayer.

Given the performance improvement that degree-heterogeneous networks have on random walk-based

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queries. It is important to highlight some other ef-fects that may increase the level of heterogeneity inreal networks, for instance remaining battery power,antenna orientation and geographical position aboveground may lead to larger cluster-heads. Also, given thedistributed nature of random walks and the push-pullalgorithm proposed, cluster-heads can rotate in orderto avoid energy depletion, and the only nodes that needto be informed are the neighbors.

Finally it is important to mention that our workdidn’t include the extra cost incurred by high-degreenodes. In wireless systems, high-degree nodes can bespecial nodes or nodes whose output power is in-creased, in both cases a high-degree node incurs ahigher cost, in the first case is an economic cost and inthe later an extra energy cost. Our work did not includethis extra cost. The cost for the case where the outputpower is increased, could be inserted in our analysis bymultiplying the extra-energy consumed by high-degreenodes by the number of high-degree nodes utilized.

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Marco Zuniga received the B.Sc degree in Electronic Engineer-ing from Pontificia Universidad Catolica, Lima, Peru in 1998. Hereceived his MS and PhD degrees in Electrical Engineering fromthe University of Southern California in 2002 and 2006. He is aPost Doctoral researcher at the University of Ireland, Galway.His current research interests are in the area of wireless sensornetworks.

Dr. Chen Avin received the B.Sc. degree in CommunicationSystems Engineering from Ben Gurion University, Israel, in 2000.He received the M.S. and Ph.D. degrees in computer science

from the University of California, Los Angeles (UCLA) in 2003and 2006 respectively. He is a Lecturer in the Department ofCommunication Systems Engineering at the Ben Gurion Uni-versity since October 2006. His current research interests are:Graphs and Networks Algorithms, Sensor Networks, RandomGraphs, Complex Systems and Random Walks.

Bhaskar Krishnamachari received the BE degree in electricalengineering from the Cooper Union, New York, in 1998 andthe MS and PhD degrees from Cornell University in 1999 and2002, respectively. He is currently an Associate Professor inthe Department of Electrical Engineering, University of South-ern California. His primary research interest is the design andanalysis of efficient mechanisms for operating wireless sensornetworks.